Actually the first minimum in an Airy pattern is seen in a close double as it is the defining valley between the spurious disks. Whether or not the CO actually improves detectability (for near equal stars) is tough to say. The numbers say it is so. Seeing however dictates such an overwhelming influence (not to mention other details like eyepieces, visual acuity, etc) that it would be tough to prove.

But if Norme says he has then so much to the good.

The WDS and other catalogues do seem to lack precision. Does anyone know a good source to well defined precise measures for a set of 'popular' doubles? It would have to include orbital elements in some cases, but thats ok.

Lewis, following Airy (much earlier), gives some of the mathematics in his 1914 paper - along with levels of relative illumination in the diffraction image, etc. Of course, this is for optics without central obstruction.

Norme, the problem of where to slice - that is, what is perceptible to the eye - is I think more of an issue than the nature of the image. EdZ has indicated one aspect of that with his comments on what is seen at varying levels of magnification. In other words - how do you decide/establish "the visual threshold" to know where to slice?

Ahh the slice quandary...

It would seem this becomes a problem not just here with doubles but in resolution limits on a number of fronts, is; planetary contrasts, lines versus disks, light features versus dark and resultant contrast. It would seem when you put the minds microscope up to drawing these finest boundaries and such it hits this ambiguous wall. One you'd normally never find if you never cut the wire that fine.

The slice issue is one of those things that seems to rest quietly till you wake it.

Pete, apparently this is one of those topics that raises it's ugly head from time to time. I wish I could find the discussion Ed mentioned a few years back. Sorry to have missed it.

Glenn, I'm really curious to know how deep I can get on a split. The challenge is part of the allure of doubles, for me, and the entire journey of discovery has been fascinating. This whole RoT exercise and trying to understand how and why unequal pairs are difficult is so very interesting, too.

I was playing around with the apparent view of 7 Tau (top) and 31 Tau, two very close, nearly equal magnitude, Dawes pairs. The pairs are very close at just over 6th magnitude (approx), too. The sketch below is not far off from actual appearance, to me. I had to make the image small to get the darker space to show accurately as I can. So, 31 Tau is reported at 0.8" (150mm Dawes at 0.8" rounded) arc and 7 Tau at 0.74" arc (0.7" rounded.) So, rounded to one decimal, there is a 0.1" arc difference, otherwise maybe only about 0.06" arc. That's an incredibly tiny amount and still showing some contrast differential (dark, but not black space.)

Still, judging from the difference between the two, I'd bet I could shave off another couple 100th's and get closer to 0.70" arc and still see some contrast fall off between them. Taking the CO into account, Dawes /should/ be closer to 0.77" arc * (1 - 0.28^2) = 0.71" arc (barring aberrations) in 0.28% obstructed 150mm aperture (which is the Sparrow limit for 150mm clear aperture.) So, to me, it seems being able to get closer is a real deal and Wilfried is onto something. A real world observation that seems to agree with the MTF. I find observing something approaching the Sparrow limit fascinating and a challenge worthy of pursuit.

My guess is, accounting for inherent aberration (making close pairs more difficult) and the slightly dimmer pairs observed (making them a tad easier), I might be able to hit about 0.72" arc.

North is left and west is bottom. Seeing was exceptional at about 9/10 on average.

Attached Files

For evidence regarding the performance of refractors and reflectors for splitting equal bright doubles I prefer again sticking to the numbers. Lord lists in his paper (to be found still under http://web.archive.o...servatory.or... ) single observations with 3 and 6" refractor and 6 and 10" reflector. While the advertised data in these reports for magnitudes is obsolete the data given for separation seems quite OK even if meanwhile WDS lists other separations due to orbits.
In relation to Dawes the 3" refractor shows for equal bright doubles one result with a 0.98 ratio and the 6" refractor shows 3 results with 0.92 ratio. For the 6" reflector Lord lists 2 results with a ratio of 0.79 and two with 0.92 and for the 10" reflector we find two results with a ratio of 0.87.
So at least this source gives evidence in favour of reflectors. There are hopefully other sources with better evidence for this topic and may be to the contrary - would be glad to know them but please do not refer to old threads that can no longer be found.
Wilfried

Quote: The WDS and other catalogues do seem to lack precision. Does anyone know a good source to well defined precise measures for a set of 'popular' doubles? It would have to include orbital elements in some cases, but thats ok.

Glenn

Glenn, the best prospect for highest accuracy measures is recent speckle measures of doubles. There are science papers, some available online, that discuss the improved level of accuracy attainable.

Otherwise, the 6th Orbit Catalog, same site as the WDS, the USNO astrometry section. Orbits with a "grade 1" rating are likely to be very accurate. Quite a few grade 2 orbits are of high standard too for getting high accuracy separations.

The WDS is a collecting house. Data will vary in accuracy, being collected from multiple sources, with varying levels of accuracy. It's the best single databasae for doubles. Of course there are errors, and the keepers of the WDS are open to being advised of these. But some data collected is of necessity going to be less accurate, or have errors.

If you look at some of the orbit plots in the 6th Orbit Catalog, especially with lower-graded orbits, say grade 3 and 4, you'll often find quite large error lines marked for some measures. It can even happen with grade 1 and 2 orbits - some data points don't fit the pattern of a large number of others so they're effectively rejected in determining the orbit.

Wilfried, I may have overlooked it, but does Lord give a description in his paper you're referring to of what he means by "resolved"? The term is, as we've seen in these forums, often used with different meanings - separated, notched, elongated, not quite round, etc.

Your graph shows a very small difference in disc size from unobstructed (1.22 units - I assume arcsecs? ) to about 1.16 units in a 30% obstructed scope.

Dave, those figures are not arc seconds, but the constant used to determine the angular size with the wavelength and the aperture. The Airy disc in a 30% obstructed scope is indeed (1.11 * 550 * .206) = 126/D radius in arc seconds - not 138/D. It's the difference between 0.92" and 0.84" for a 150mm clear and obstructed aperture, respectively.

The additional diffraction effects can be approximated using (1 - co^2.) So, in an obstructed aperture, the Airy disc is smaller with added diffraction by a factor of (1 - co^2), but also dimmer and smaller due to both diffraction and obscuration by a factor of (1 - co^2)^2 normalized to 1. It is the reason a 30% obstructed scope puts 68% of the light into the disc and 32% into the rings: the rings are brighter and the disc is dimmer and smaller.

Now, is that significant? In good seeing, cooled, perfectly collimated, and with a reasonably good Strehl (=/> 0.95) I am convinced (real world observation) it is.

It allows an obstructed scope to resolve closer pairs and still maintain some contrast between peaks. But, it also hampers unequal pairs with greater separations, complicated by the fact the second ring is a full magnitude dimmer in a moderately obstructed scope just as Treanor's chart above shows. The chart also shows how the disc shrinks in angular size at 50% intensity, and Treanor thought it to be significant.

Wilfried, I may have overlooked it, but does Lord give a description in his paper you're referring to of what he means by "resolved"? The term is, as we've seen in these forums, often used with different meanings - separated, notched, elongated, not quite round, etc.

Certainly a good question - Lord refers basically to Dawes, but also to Rayleigh, Sparrow etc. but does not give any definition what is to be considered as resolved. As Lord is using historical material from the Lewis collection of observations this question cannot be answered with any confidence as there are too many people involved - we can only trust in the seriosity of the observers.
Interesting also the last page of his paper I have overlooked until now, would have spared me some own calculations: The statement "Resolution of equal binaries may be marginally improved by introducing a central obstruction" is on top of a table with a calculation of the influence of a CO on the Rayleigh criterion.
Wilfried

Norme, if I see pairs such as in your illustrations I call them a split and move on.

I use Dawes and Rayleigh to select pairs for viewing - my new 8" TEC Mak-Cass will be examining lots of pairs from 0.4" to 1.0" because it's theoretical limit for splitting/resolving/dividing close pairs is around 0.6"...

Dave, I fell into some great seeing during retirement, something not common during my professional career, and often so very thankful. It has allowed me to push the limits in ways I've never imagined. Without getting too religious, sometimes I look up and just thank God for the blessing. Few things are more beautiful than a well behaved Airy pattern.

Ed, having read some of the posts...I see where you mention more than once about the Airy disc size. In fact, "Although it could be a lifelong study, I wonder just at what point or how abruptly a transition from 'smaller disks' to 'more difficult to see' does occur? These thoughts need to go in the other thread when I get time." If you had not done so, now is the time.

I wish I could find that article, this seems to be discussed over and again. Anyway, Wilfried is looking for a way to figure the visible disc size, what would be the best way to attack that on paper, first? I used what I understood your article on resolution to mean: 50% Airy disc diameter and extrapolating down to limiting magnitude. Maybe that is not exactly the right method.

Fred mentioned slicing the PSF is too subjective or would be for different observers, but it might be a start.

I think part of the reason the slicing of the pie so to speak or lined boundary between value and another creates a quandary is that no line or slice can exist. Between values it would seem the lines, slices or limits are soft edged with a null zone of sorts. Within this grey area you could put varying perception between different observers and such so this zone is a grey area of averages.

You may be right, Pete, it's a complicated topic. Maybe Wilfried can just use the 50% figures in the chart he posted above. Maybe Treanor ran into the same problem and simply chose a cut off. But the figures are still a tiny bit smaller suggesting a tiny bit more resolution.

One thing becoming somewhat apparent to me is, when the conditions warrant - when all induced aberrations are minimal, those scales can make a tiny difference. You can push just a little deeper, maybe to the point where the size of the spurious disc will either make or break a tight split.

For example, I would never think of calling a split on 72 Pegasi at 0.56" arc. I called one on 7 Tau at 0.74" arc. And there is the slightest hint of a possible, barely detectable, tiniest hint (you get the point) of a very faint dark space on STT 517 at 0.67" arc. Not enough to call a split, but just enough to make you wonder. That seems unprecedented in theory and in practice.

So, there is probably no definite point where a split becomes non resolution, except for the Sparrow limit in concept, in theory, and maybe in practice. But, what is the Sparrow limit, 107/D is all scopes? There is probably a more gradual change up to the point where contrast is truly flat across the peaks. So, what is that point?

Point being, I think Wilfried asked a fascinating question that could use a good answer. I wish I had one for him. I do think about it a lot, and I do push my scope to those limits to see.

Maybe I do. I'd have to dig up and verify my work, but at one point I calculated my own scope to have a high frequency resolution similar to a 6.4" clear aperture. Then, at lower frequencies larger than the first ring, I calculated it's mid range contrast transfer to be close to a 4.3" clear aperture. (I think I even included some aberration, Strehl 0.95 for my scope and assumed a Strehl 0.98 for the clear aperture.)

Now, since unequal pairs with separations within the frequency range corresponding to the first three or four rings seem to behave like planetary contrast at the same scale, so this treatment might be applicable to unequal pairs, too. After all, it is diffraction contrast that makes either difficult.

So, for tight pairs, treat an obstructed scope just like a slightly larger clear aperture. Conversely, for mid range frequencies out to about 3x or 4x Raleigh, treat an obstructed scope like a smaller aperture. A good figure can be calculated, for perfect or aberrant apertures, and the size of the spurious disc might be implicit in the answer.

Ed, thanks for the posted links - I will go through them in the next days and will probably take the liberty to add some of the reported splits to my so far rather small data set of limit observations for my RoT project. Wilfried

Glenn, the best prospect for highest accuracy measures is recent speckle measures of doubles. There are science papers, some available online, that discuss the improved level of accuracy attainable.

Otherwise, the 6th Orbit Catalog, same site as the WDS, the USNO astrometry section. Orbits with a "grade 1" rating are likely to be very accurate. Quite a few grade 2 orbits are of high standard too for getting high accuracy separations.

Thanks Fred! You jogged my memory into recalling the orbit grades. Interestingly Castor has a grade of '3' which was one I had interest in as I captured it as a test two years ago in computing plate scale. I realize a wider separation would be beneficial, but most binaries with good grades are very close doubles.

As far as the size of the spurious disk, one can calculate that assuming it is defined as something like full width half max. Using the wave eqn solution (no aberratons) with a possible obstruction, its not difficult to do. And it appears further up in this thread someone already did that calculation.

However the visual appearance I believe would be larger than that. If you were to use an insane amount of magnification, and had seeing of the gods, I suspect you would measure it out to something closer to the intensity of the first bright ring, or closer to 20% of the central max. But the flaw there is that the perception would be quite sensitive to the magnitude of the star. Bright stars would 'measure' quite a bit larger than faint stars.

Glenn, my thoughts were to forget the spurious disc and simply go with line pairs. Calculate the maximum line pairs at either Dawes or Abbe limit, as modified by (1 - co^2), then calculate the clear aperture required to attain that spacial frequency. That clear aperture and the obstructed aperture should have the same maximum spacial frequency. So, you could treat resolution of close pairs the same way applying different apertures.

You can even apply aberration to the calculation using very good Strehl of 0.98 for the refractor and 0.95 for the obstructed scope. Using line pairs implies the spurious disc size. That would work for maximum spacial frequencies, in terms of getting an figure.

I don't remember how to treat lower frequencies. I may have determined the aberrant contrast transfer for both at lower frequencies, then applied the (D - co) rule of thumb to come up with a percentage of the clear aperture required. Its basically what we do we we say a 150mm - 42mm = 109mm clear aperture, then apply the a reasonable Strehl to both (instead of referring to "perfect" apertures like MTF does.) This should be more realistic, despite the non linear eyeball function.

Somehow you have to work in the lessor peak intensity of the obstructed scope (causing more light in the rings) using a approximation of (1 - co^2)^2 to determine their Strehl-like number and effective contrast transfer. The clear aperture definitely has the advantage here, so it can be smaller and perform as well.

Measuring them at insane magnifications would work, too. Dave mentioned that above. He seemed disinclined to do so because of the difficulties in getting 800x on a good night. He may have a point, and it would require calibration and some special equipment. It might be subject to errors, too. It could be done, but it might be done on paper, as well.

According to the topic of this thread all observation reports were with may be one exception for equal doubles - certainly all interesting regarding the size of the spurious disk but to my regret of no use for the RoT project.Wilfried

Back to the topic of this thread - the size of the spurious disk. Rereading the strong arguments of Taylor for the use of reflectors for double star observing I found no reason to doubt his measurement even if some of his reported observations shows extreme Dawes ratios down to 0.5 (without considering the effects of CO - Taylor mentions the use of artificial high COs for resolving extremely tight doubles).At the same time I do not see any reason to question the observations of EdZ.Referring to the numbers given for the size of the spurious disk in the already posted table of Chris Lord I get a relation of 42% of the Airy disk for non obstructed scopes. While Lord explains very well how he calculated the other numbers in his table he did this regrettable not for his numbers for the size of the spurious disk.But may be the size of the spurious disk lies in the eye of the observer?Wilfried